1. TWO-WAY SLAB
INTRODUCTION
Slab panels which deforms in two orthogonal directions must be designed as “Two-Way Slabs”, with the principal
reinforcement in the two directions.
Two-Way Slabs may be of the following types:
1 Slab Supported on Wall or Simply Supported Slabs
(i) Simply Supported with corners to lift
(ii) Simply Supported with corners held down or not free to lift (Restrained Slabs)
2 Continuous Slabs
3 Framed Structure – Beams-Slab
One-Way and Two-Way Action of Slab
The One-Way Slab action may be assumed when the predominant mode of flexure in one direction only.
Rectangular slabs which are supported only on two opposite sides by unyielding supports (Walls) and are
uniformly loaded (along the direction parallel to the supports) is an ideal example of One-Way Slab action as
shown in Fig 1 (a).
The bending is in one direction only.
1
2. The primary one-way action ceases to exist if either the support conditions or loading conditions are altered.
For example, if the uniformly loaded rectangular slab is supported on all four edges then the deformed
surface of the of the slab will be “Doubly Curved”, with the load being transferred to all the four supporting
edges [Fig 1(b)].
Such action is called a Two-Way action involving significant curvatures along two orthogonal directions.
The typical variations of longitudinal and transverse bending moments are depicted in Fig 1 (c).
The bending moments are maximum at middle of the slab.
The Moment MX along the short span (LX) is invariably greater than MY along the larger span (LY)
As the aspect ratio LY/ LX i.e. (long span/short span) increases, the curvatures and moments along the long
span progressively reduces, and more and more of the slab load is transferred to the two long supporting
edges by bending in the short span direction.
In such cases the Bending Moment MY is generally low in magnitude [Fig 1(d)].
Hence such longer rectangular slabs [LY/ LX > 2 ] may be approximated as One-Way Slab for convenience in
analysis and design.
2
5. Reinforcement :- In a two way slab the reinforcement is provided along the transverse and longitudinal
directions.
Now if
(i) LY/ LX = 1 i.e. LX = LY
Then it’s a Square Two Way Slab
(ii) LY/ LX > 2
Then it’s Rectangular Two Way Slab
Difference Between Wall-Supported Slabs and Beam / Column Supported Slabs
The distributed load ‘W’ on a typical Two-Way Slab is transmitted partly as (WX) along the short span to the
long edge supports, and partly as (WY) along the long span to the short span to the short edge supports.
In wall supported panels, these load portions i.e. (WX, WY) of the load are transmitted by the respective wall
supports directly to foundation as shown in Fig. 2 (a).
On the contrary when the edge supports to the slab consists of Beams spanning between columns, then the
portion of the load is transferred to the beams and consequently the beams transmits the load in the
perpendicular direction to the two supporting columns and from columns to the foundation as shown in
Fig. 2 (b).
LY
LX
LX
LY
5
6. Fig 2 Load Transfer in Wall-Supported and
Column-Supported Slabs
6
7. Slab Thickness Based on Deflection Control Criterion
The initial proportioning of the slab thickness may be considered by adopting the same guidelines
regarding span/effective depth ratios as applicable to One-Way Slab.
The shorter span LX is taken to be the effective span
The percentage of the tension reinforcement required in the short span direction in a Two-Way slab is
generally less than what is required for a One-Way Slab for the same effective span.
Hence the Modification Factor to be considered for Two-Way Slab may be taken to be higher than that
is recommended for One-Way Slab. A value of 1.5 may be taken as Modification Factor for preliminary
design.
The effective depth thus provided should be verified base on the actual percentage of steel pt provided.
For a special case of Two-Way Slab with Spans up to 3.5 m and live loads not exceeding 3.0 kN/m2, the
IS Code 456 – 2000 [Clause 24.1, note 2; page 39] permits the slab thickness (Overall depth D) to be
calculated directly as follows, without the need for subsequent checks on deflection control:
7
8. “For Two-Way Slabs of shorter Spans (up to 3.5 m) with mild steel Reinforcement, the Span to Overall
Depth Ratios given below may generally be assumed to satisfy vertical deflection limits for loading class
up to 3.0 kN/m2.
Simply Supported Slabs = 35
Continuous Slabs = 40
For high strength deformed bars of grade Fe 415, the values given above should be multiplied by 0.8.
OR
(i) Using Mild Steel ( Fe 250 Grade)
D ≥ (1)
(ii) Using Fe 415 Grade Steel
D ≥ (2)
LX/35 For Simply Supported Slab
LX/40 For Continuous Slab
LX/28 For Simply Supported Slab
LX/32 For Continuous Slab
8
9. METHODS OF ANALYSIS
Two-Way Slabs are highly statically indeterminate structural elements.
They may be visualized as comprising of intersecting, closely spaced beam strips which are subjected to
Flexure, Torsion and Shear.
Owing to the high static indeterminacy rigorous solutions are generally not available.
In order to accommodate the differences observed experimentally due to Non-Homogeneous and Non-
Linear behavior of concrete, Wester-guard proposed solutions in the form of convenient Moment
Coefficients.
Such Coefficients have been widely used by codes all over the world.
Another approximate method, which is very elementary in approach is the Rankin-Grashoff Method.
This method simplifies the highly indeterminate problem to an equivalent simple determinate one.
The modern computer-based methods included “Finite Difference Method” and “Finite Element
Method”.
Other methods used for Limit State Design are “Inelastic Methods” base on Yield Line Analysis.
9
10. SLABS SPANNING IN TWO DIRECTIONS AT RIGHT ANGLES
According to IS 456 – 2000, clause 24.4 page 41
“The Slabs Spanning in Two Directions at right angles and carrying uniformly distributed load may be
designed by any acceptable theory or by using coefficients given in Annexure D; page 90
Note:- The most commonly used elastic methods are based on Pigeaud’s or Wester-guard’s Theory and the
most commonly used Limit State of Collapse Method is based on Johnson’s Yield – Line Theory.
In the case of uniformly loaded Two-Way Rectangular Slabs, the IS Code suggests the following
design procedure for:
RESTRAINED SLABS: (Annexure D – 1 , page 90)
When the corners of a slab are prevented from lifting, the slab may be designed as per the following steps:
1 The Maximum Bending Moments per unit width in a slab are given by the following equations:
MX = αX. W L2
X
MY = αY. W L2
X
10
11. Where:
αX and αY are coefficients given in Table 26 page 91 of IS 456 – 2000
W = Total Design Load per unit area
MX, MY = Moments on Strips of unit width spanning LX and LY respectively
LX and LY = Lengths of Shorter and Longer Spans respectively
2 Slabs are considered as divided in each direction into middle strips and edge strips as shown in Fig 3,
the middle strip being three-quarters of the width and each Edge strip as One Eight of the width.
3 The maximum moments calculated as in 1 apply only to the middle strips and no redistribution shall be
made.
4 Tension Reinforcement provided at Mid-Span in the middle strip shall extend in the lower part of the
slab to within 0.25 L of a continuous edge, or 0.15 L of a discontinuous edge.
5 Over the continuous edges of middle strip, the tension reinforcement shall extend in the upper part of
the slab to a distance of 0.15 L from support, and at least 50% shall extend to a distance of 0.3 L
6 At a discontinuous edge, negative moments may arise. They depends on the degree of fixity at the edge
of the slab, but in general, tension reinforcement equal to 50% of that provided at mid span extending to
0.1 L into the span will be sufficient.
11
13. 7 Reinforcement in Edge Strip, parallel to that edge shall comply with the minimum given in section 3 of
the code [Cl. 26.5.2.1 pp. 48].
8 Torsion reinforcement shall be provided at any corner where the slab is simply supported on both edges
meeting at the corner. It shall consist of Top and Bottom reinforcement, each with layers of bars placed
parallel to the sides of the slab and extending from the edges a minimum distance of 1/5th of the shorter
span. The area of reinforcement in each of these four layers shall be three-quarters of the area required
for the maximum mid-span moment in the slab.
9 Torsion reinforcement equal to half that described in point 8 shall be provided at a corner contained by
edges over only one of which the slab is continuous.
10 Torsion reinforcement need not be provided at any corner contained by edges over both of which the
slab is continuous.
11 When LX / LY > 2, the slabs shall be designed as spanning One-Way.
13
14. SIMPLY SUPPORTED SLABS: (Annexure D – 2 , page 90)
1 When simply supported slabs do not have adequate provisions to resist torsion at corners and to
prevent the corners from lifting, the maximum moments per unit width are given by the following
equations:
MX = αX. W L2
X
MY = αY. W L2
X
Where,
MX, MY, W, LX , LY are the same as for restrained slabs
αX and αY are moment coefficients given in Table 27 page 91 of IS 456 – 2000
2 At least 50% of the tension reinforcement provided at mid-span should extend to the supports.
3 The remaining 50% should extend to 0.1 LX or 0.1 LY of the support, as appropriate.
¾ LX
LY
LX
LX/8
¾ LY
LY
LX
LY/8 LY/8
LX/8
Fig 3 Division of Slab into Middle and Edge Strip 14
15. Uniformly Loaded and Simply Supported Rectangular Slabs
The moment coefficients prescribed in the code clause D – 2 Page 90 to determine the maximum moments
per unit width in the short span and long span directions are based on the Rankin-Grashoff theory
According to this theory, the slab can be divided into orthogonal strips of unit width (Beams).
The load is so proportioned to long and short span strips such that the deflections δ of the two middle
strips at their crossing is same [Fig. 4].
Fig. 4
15
16. If torsion between the interconnecting strips and the influence of adjoining strips is ignored, each of the
two middle strips can be considered to be simply supported and subjected to uniformly distributed load
WX (On the Short Span Strip) and WY (On the Long Span Strip) [Fig 4].
Hence the mid-point deflection δ can be obtained as :
𝛅 =
𝟓
𝟑𝟖𝟒
𝐖𝐗 𝐋𝐗
𝟒
𝐄𝐈
=
𝟓
𝟑𝟖𝟒
𝐖𝐘 𝐋𝐘
𝟒
𝐄𝐈
(3)
where
I is same for both the strips
Hence a simple relation between WX and WY is obtained as
WX = WY (LY / LX)4 (4)
Also
WX + WY = W (5)
Putting value of WX from eq. 4 in eq. 5
WY (LY / LX)4 + WY = W
16
17. Or,
W = WY [ 1 + (LY / LX)4 ]
Let
LY / LX = r
W = WY [ 1 + r4 ]
Or
𝐖𝐘 =
𝐖
𝟏 + 𝐫𝟒 (6)
From eq. 4 and taking LY / LX = r
𝐖𝐘 =
𝐖 𝐗
𝐫𝟒 (7)
Putting value of WY from eq. 7 in eq. 5
𝐖𝐗 +
𝐖𝐗
𝐫𝟒 = 𝐖
OR
𝐖𝐗 =
𝐖 𝒓𝟒
𝟏 + 𝐫𝟒 (8)
17
18. Where;
r : Aspect Ratio = LY/LX
When,
r = 1 LX = LY
Then
The slab is a Square Slab
And
𝐖𝐗 =
𝐖 × 𝟏𝟒
𝟏 + 𝟏𝟒 =
𝐖
𝟐
𝐖𝐘 =
𝐖
𝟏 + 𝟏𝟒 =
𝐖
𝟐
When
r = 2
Then
The slab is a Rectangular Slab
LX
LY
18
19. And
𝐖𝐗 =
𝐖 × 𝟐𝟒
𝟏 + 𝟐𝟒 =
𝟏𝟔𝐖
𝟏𝟕
= 0.94 W kN
𝐖𝐘 =
𝐖
𝟏 + 𝟐𝟒 =
𝐖
𝟏𝟕
= 0.06 W kN
The load contribution in larger span is much lesser than the load in the shorter span.
Hence the Moment contribution will also be in the same proportion.
Now,
𝐌𝐗 =
𝐖𝐗 𝐋𝐗
𝟐
𝟖
(9)
𝐌𝐘 =
𝐖𝐘 𝐋𝐘
𝟐
𝟖
(10)
Now substituting the values of WX and WY from Eq. 6 and Eq. 8 in Eqs. 9 and 10 respectively.
𝐌𝐗 =
𝐖 𝒓𝟒
𝟏 + 𝐫𝟒 ×
𝐋𝐗
𝟐
𝟖
𝐌𝐗 = 𝐖 𝐋𝐗
𝟐 𝟏
𝟖
×
𝐫𝟒
𝟏 + 𝐫𝟒
19
21. OR
𝐌𝐘 = 𝛂𝐘 . 𝐖 . 𝐋𝐗
𝟐
Where
𝛂𝐘 =
𝟏
𝟖
𝐫𝟐
𝟏 + 𝐫𝟒
The equations MX and MY are both in terms of short span LX.
These equations are same as given in IS 456 – 2000 [Cl. D-2 ; page 91]
The values of moment coefficients αX and αY for different aspect ratios r = LY / LX are given in
Table 27 of IS 456 – 2000; page 91 for slabs simply supported on four sides.
For other boundary conditions table 26 shall be used
Now
𝜶𝑿 =
𝟏
𝟖
𝒓𝟒
𝟏 + 𝒓𝟒 And
𝜶𝒀 =
𝟏
𝟖
𝒓𝟐
𝟏 + 𝒓𝟒
21
22. The short span moment coefficient progressively increases and the long span moment coefficient αY decreases
progressively as the Aspect Ratio r increases.
In the case of a square slab:
r = 1
αY = 0.0625
For higher values of aspect ratio r, where αX approaches the value of 1/8 i.e.
𝐫𝟒
𝟏+ 𝐫𝟒 ≈ 𝟏, then αY becomes negligible.
e.g.
Take r = 5
𝜶𝑿 =
𝟏
𝟖
𝟓𝟒
𝟏 + 𝟓𝟒 =
𝟏
𝟖
𝟔𝟐𝟓
𝟏 + 𝟔𝟐𝟓
= 1/8 × 0.998
OR
αX = 0.125
And
𝜶𝒀 =
𝟏
𝟖
𝟓𝟐
𝟏 + 𝟓𝟒 =
𝟏
𝟖
×
𝟐𝟓
𝟔𝟐𝟔
= 1/8 × 0.04
OR
αY = 0.005 Almost negligible
Hence the moment in the long span direction is almost zero, meaning thereby that the action is now a One-Way
action
22
![ The primary one-way action ceases to exist if either the support conditions or loading conditions are altered.
For example, if the uniformly loaded rectangular slab is supported on all four edges then the deformed
surface of the of the slab will be “Doubly Curved”, with the load being transferred to all the four supporting
edges [Fig 1(b)].
Such action is called a Two-Way action involving significant curvatures along two orthogonal directions.
The typical variations of longitudinal and transverse bending moments are depicted in Fig 1 (c).
The bending moments are maximum at middle of the slab.
The Moment MX along the short span (LX) is invariably greater than MY along the larger span (LY)
As the aspect ratio LY/ LX i.e. (long span/short span) increases, the curvatures and moments along the long
span progressively reduces, and more and more of the slab load is transferred to the two long supporting
edges by bending in the short span direction.
In such cases the Bending Moment MY is generally low in magnitude [Fig 1(d)].
Hence such longer rectangular slabs [LY/ LX > 2 ] may be approximated as One-Way Slab for convenience in
analysis and design.
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